Editing 24-cell (section)

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a ''[[Rectification (geometry)|rectified]] 16-cell'', with B<sub>4</sub> or [3,3,4] symmetry drawn bicolored with 8 and 16 [[octahedron|octahedral]] cells. Lastly it can be constructed from D<sub>4</sub> or [3<sup>1,1,1</sup>] symmetry, and drawn tricolored with 8 octahedra each.<!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ -->

{| class="wikitable collapsible collapsed"
!colspan=12| Three [[Net (polytope)|nets]] of the ''24-cell'' with cells colored by D<sub>4</sub>, B<sub>4</sub>, and F<sub>4</sub> symmetry
|-
![[Rectified demitesseract]]
![[Rectified 16-cell]]
!Regular 24-cell
|-
!D<sub>4</sub>, [3<sup>1,1,1</sup>], order 192
!B<sub>4</sub>, [3,3,4], order 384
!F<sub>4</sub>, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[Octahedron|octahedral]] cells.
|width=213|One set of 24 [[Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[Vertex figure]]'''<br>(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[regular complex polygon]] <sub>4</sub>{3}<sub>4</sub>, {{CDD|4node_1|3|4node}} or {{CDD|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is <sub>4</sub>[3]<sub>4</sub>, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope <sub>3</sub>{4}<sub>3</sub>, {{CDD|3node_1|4|3node}} or {{CDD|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. <sub>3</sub>{4}<sub>3</sub> has 24 vertices, and 24 3-edges. Its symmetry is <sub>3</sub>[4]<sub>3</sub>, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{CDD|node_1|3|node|4|node|3|node}}
!<sub>4</sub>{3}<sub>4</sub>, {{CDD|4node_1|3|4node}}
!<sub>3</sub>{4}<sub>3</sub>, {{CDD|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{CDD|node|3|node|4|node|3|node}}, order 1152
!<sub>4</sub>[3]<sub>4</sub>, {{CDD|4node|3|4node}}, order 96
!<sub>3</sub>[4]<sub>3</sub>, {{CDD|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]<BR>24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]<BR><sub>4</sub>{3}<sub>4</sub>, {{CDD|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]<BR><sub>3</sub>{4}<sub>3</sub> or {{CDD|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[uniform 4-polytope]]s can be derived from the 24-cell via [[Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[bitruncated 24-cell]], which is [[cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[golden ratio]] to produce the 96 vertices of the [[snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[octahedron]] produces an [[Regular icosahedron|icosahedron]], or "[[Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]."

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[polygon]] nor a [[simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[great 120-cell]] and [[grand stellated 120-cell]]. With itself, it can form a [[polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}